Spacetime noncommutativity also arises by considering an electron in a strong magnetic field \(B\) . In
this regime, due to the minimal coupling with the background gauge
field \((A_x, A_y) = (0, Bx)\)  associated to the flux \(B\) , the dynamics
takes place in the reduced phase space \(q = y, p = Bx\) . Thus the
electron’s coordinates become noncommutative: \([x, y] =-\frac{i\hbar }{B}\)  (cf. [1]} for an extended discussion).
Hence quantum theory in the presence of a magnetic field \(B\)  leads to a noncommutative spacetime.
Similarly, low energy effective actions of open strings in the
presence of a background Neveu-Schwarz \(B\) -field can be described by
gauge theories on noncommutative spaces.
The study of Yang-Mills (and Born-Infeld) theories on noncommutative spaces has proven very fruitful:
it allows to realize string theory T-duality symmetry within the low
energy physics of noncommutative (super) Yang-Mills theories [2]}
It provides exact low energy D-brane effective actions, in a
given \(\alpha ^{\prime }\rightarrow 0\)  sector of string theory where closed strings
decouple, see ref. [3]}. In that paper
Seiberg and Witten provided an
explicit map (change of variables) between commutative and
noncommutative gauge theories.
